﻿ 新型冠状病毒肺炎疫情多次暴发的动力学机制分析
 中华流行病学杂志  2021, Vol. 42 Issue (6): 966-976 PDF
http://dx.doi.org/10.3760/cma.j.cn112338-20210219-00123

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#### 文章信息

Xiao Yanni, Li Qian, Zhou Weike, Peng Zhihang, Tang Sanyi

Analysis on dynamical mechanism of multi outbreaks of COVID-19

Chinese Journal of Epidemiology, 2021, 42(6): 966-976
http://dx.doi.org/10.3760/cma.j.cn112338-20210219-00123

### 文章历史

1. 西安交通大学数学与统计学院/数学与生命科学交叉中心 710049;
2. 陕西师范大学数学与统计学院, 西安 710019;
3. 南京医科大学公共卫生学院 211166

Analysis on dynamical mechanism of multi outbreaks of COVID-19
Xiao Yanni1 , Li Qian1 , Zhou Weike2 , Peng Zhihang3 , Tang Sanyi2
1. Center for Intersection of Mathematics and Life Sciences, School of Mathematics and Statistics, Xi'an Jiaotong University, Xi'an 710049, China;
2. School of Mathematics and Statistics, Shaanxi Normal University, Xi'an 710019, China;
3. School of Public Health, Nanjing Medical University, Nanjing 211166, China
Abstract: Objective In the context of COVID-19 pandemic, the epidemic severities, non-pharmaceutical intervention intensities, individual behavior patterns and vaccination coverage vary with countries in the world. China has experienced a long period without indigenous cases, unfortunately, multi local outbreaks caused by imported cases and other factors have been reported, posing great challenges to COVID-19 prevention and control in China. Thus it is necessary to explore the mechanisms of the re-emerged COVID-19 epidemics and their differences. Methods Based on susceptible exposed infectious recovered (SEIR) epidemic dynamics model, we developed a set of novel evolution equations which can describe the dynamic processes of integrated influence of interventions, vaccination coverage and individual behavior changes on the re-emergency of COVID-19 epidemic. We developed methods to calculate the optimal intervention intensity and vaccination rate at which the size of susceptible population can be reduced to less than threshold for the re-emergency of COVID-19 epidemic. Results If strong interventions or super interventions are lifted too early, even a small cause can lead to the re-emergence of COVID-19 epidemic at different degrees. Moreover, the stronger the early control measures lifted are, the more severe the epidemic is. The individual behavior changes for the susceptibility to the epidemic and the enhancement or lifting of prevention and control measures are key factors to influence the incidence the multi outbreaks of COVID-19. The optimist early intervention measures and timely optimization of vaccination can not only prevent the re-emergency of COVID-19 epidemic, but also effectively lower the peak of the first wave of the epidemic and delay its arrival. Conclusion The study revealed that factors for the re-emergence of COVID-19 epidemics included the intensity and lifting of interventions, the change of individual behavior to the response of the epidemic, external incentives and the transmissibility of COVID-19.
Key words: COVID-19    Outbreak    Intervention    Susceptible population    Model of transmission dynamics

 注：大连市两次暴发时间分别为2020年7月22日至8月5日、12月15日至2021年1月6日 图 1 我国新型冠状病毒肺炎第一波疫情清零后的部分地区疫情与二次暴发疫情

1.模型构建:融合干预措施以及由感染者驱动的个体行为改变和疫苗接种等多因素，结合经典的SEIR模型，推导本文需要采用的数学模型[9]。考虑人群面对新发或重大传染病时，将自发改变其行为以减小感染风险，易感人群(S)分为两类：行为正常的易感人群和行为发生改变的易感人群。由于易感者被感染后关联的潜伏者未出现症状，因此，潜伏者与易感者视为行为相同，将其分为行为正常的潜伏人群和行为发生改变的潜伏人群。经典的SEIR疾病传播动力学模型公式：

(*)

(**)

(***)

p=(Sn+En)/(S+E)表示易感者和潜伏者中保持正常行为的人的比例，其中S=Sn+SaE=En+Ea，则当时，模型(***)可近似为：

(1)

2.有效再生数、感染规模和数值分析:当所有的易感个体在疾病暴发期间行为不发生任何变化，即p=1时，根据经典的SEIR模型的主要结论得到相应的基本再生数为：

(2)

 注：经典的SEIR模型在给定人口规模为1万时，参数f=0，p=1和ν=0时；A.基本再生数R0分别取值为1.5、2、3时，最终感染规模和易感人群阈值Sth0的关系；B.相应的有效再生数；C和D.基本再生数R0=3以及p=1时，取f = 0、0.3、0.450 9、0.5和0.6分别表示无、弱、最优、强和超强干预措施后的数值实现，其中fc=0.450 9，Sth0=3 333.3，在T=400 d完全解除干预 图 2 经典的SEIR模型在完全自然条件下的解和有效再生数的数值实现

1.最优的早期干预措施强度与二次暴发风险:首先分析仅有强化干预措施时，如何设计参数使得新冠肺炎疫情得到最佳的防控效果(易感人群规模达到其阈值)。为此，假设行为改变率p=1和v=1，此时、和。因此，为了寻求最佳的干预强度，假设存在一个干预强度的临界值，使得当干预强度取值为fc时，有干预措施后易感人群的稳定水平S，恰好达到无干预措施时的易感人群规模阈值Sth0，根据公式(2)得到公式：

(3)

2.个体行为动态演化与二次暴发风险:模型(1)中的个体行为改变率p在疫情的驱动下满足一个微分方程，而反过来个体的行为变化又降低传染的发生率。由此可见感染者的数量与易感者、潜伏类的行为变化之间的相互作用关系，必将导致新冠肺炎疫情发生复杂的动态演化，这为揭示常态化防控策略下的疫情演化具有实际指导意义。为分析上述关系对新冠肺炎疫情的影响，设定人口规模为1万时，固定基本再生数R0=4以及f=0和v= 0，并取m = 0.5，0.1，0.05，0.02和0.01分别表示易感者和潜伏类个体对疫情的超强敏感、强敏感、敏感、弱敏感和不敏感，此时Sth0=2 500。见图 3

 注：设定人口规模为1万时，固定基本再生数R0 =4以及f=0和v = 0，并取m = 0.5、0.1、0.05、0.02和0.01分别表示易感者和潜伏类个体对感染数量的超强敏感、强敏感、敏感、弱敏感和不敏感时的数值实现，此时Sth0 =2 500 图 3 经典的SEIR模型在易感者和潜伏类个体对感染数量的不同敏感程度的数值实现

3.疫苗接种与优化常态化防控:前两节分析的是解除强干预措施以及个体行为改变对感染者的敏感程度是诱导疫情多次暴发的重要因素。为了恢复正常生活，强干预措施不可能始终实施，个体也由于防控意识疲劳、工作所需等也不可能对疫情长期处于高度敏感状态。因此，疫苗的研发成功和有效接种是弥补上述缺陷的关键。

(4)

 注：设定人口规模为1万，固定基本再生数R0=3以及p=1，并取f=0.5和0.6分别表示强和超强干预措施后优化接种疫苗的数值实现；A和B. f=0.5，v=0、0.01和vc=0.032 6；f=0.6，v=0、0.01、0.05和vc=0.126 8；C和D. f=0，m=0.02（0.05），v=0、0.001、0.002、0.005（0.003） 图 4 经典的SEIR模型在强和超强干预措施后优化接种疫苗的数值实现

f=0.5时，通过计算得到Sth0=3333.3，ST0 = 4188.8，vc=0.0326，即当干预措施的强度为强干预时，在解除干预措施之前接种1周的疫苗，且接种率为0.0326就能避免二次暴发（图 4AB的灰色曲线）；当f=0.6时，计算得到Sth0=3333.3，ST0=8096.4，vc=0.1268，即当干预措施的强度为超强干预时，在解除之前接种1周的疫苗，且接种率需要提升到0.1268才能实现目标（图 4AB的红色曲线）。因此，实施防控措施越强的地区，在解除防控措施之前，为避免更大的二次暴发，就必须让疫苗的接种比例更大。

4.强干预措施下疫情诱导的行为改变与疫情的多次暴发:考虑干预措施、民众行为动态变化与免疫接种3个策略的协同优化效应对新冠肺炎疫情的影响。固定参数值见图 5，考虑民众对疫情相同敏感性下干预强度与接种率对疫情多次暴发的影响。比较图 5BDFH的红色曲线，发现：①干预措施持续存在时，相对较弱的干预措施会导致相对较强的第一波疫情、较弱的第二波疫情，且第一波疫情随着干预措施的加强不断减弱；②干预措施解除之后，原来较弱的干预措施会产生相对较弱的第三波疫情，且第三波疫情随着原来干预措施的加强不断增强。为分析疫苗接种与解除干预措施对疫情影响的协调作用，在解除干预措施的前1周提高了接种率，发现相同的接种率对于补偿干预措施解除的效果差异很大(图 5中蓝色曲线)。因此，对于相对较弱的防控措施(f = 0.3)，解除其之前1周增加接种率v=1，能够有效抑制疫情的二次暴发。这是由于解除措施时易感人群规模相对较低(图 5B)。然而，解除相对较强的干预措施，再增加同样的接种率的情况下，尽管不能抑制下一波疫情的暴发，但能避免后期的反复多波疫情的发生。见图 5DFH

 注：设定人口规模为1万，固定基本再生数R0 =6，f=0.3、0.4、0.5和0.6，同时考虑个体行为变化与疫苗接种等多因素协同效应时的数值实现。同样在第400天解除干预措施，其中f≠0和v=0.001对应的是红色曲线，f=0和v=0.01对应的是蓝色曲线 图 5 经典的SEIR模型同时考虑个体行为变化与疫苗接种等多因素协同效应时的数值实现

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